PHYSICS OF FLUIDS 17, 021703 共2005兲

Mixing induced by Rayleigh–Taylor instability in a vortex Laure Coquart,a兲 Denis Sipp, and Laurent Jacquin ONERA, 8 rue des Vertugadins, 92190 Meudon, France

共Received 28 October 2004; accepted 7 December 2004; published online 19 January 2005兲 The direct numerical simulation 共DNS兲 of a two-dimensional Lamb–Oseen vortex with a heavy internal core has been performed. Linear stability theory predicts the existence of Rayleigh–Taylor 共RT兲 instabilities due to the destabilizing effect of the centrifugal force on the radial flow nonhomogeneities. The DNS first exhibits wavy azimuthal perturbations which are nonlinearly distorted into bubble-like patterns, characteristic of the standard development of the RT instabilities, i.e., instabilities obtained in a planar nonhomogeneous flow in the presence of gravity. Nevertheless, important differences may be observed in the late stage development of the instability: contrary to the standard case, the bubbles are then stretched in the azimuthal direction leading to a strong radial filamentation of the flow. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1852580兴 Stabilizing mechanisms associated to rotation usually make a vortex very resistant to radial momentum diffusion.1,2 The present Letter is considering possible density variation effects to achieve this goal. If density effects were significant, vortex control by means of injection of heated or cooled air could be considered, for example, in the application of aircraft wakes.3 The goal of this Letter is, precisely, to evaluate the potential of such density effects on the dynamics of vortex cores. The linear stability of a compressible two-dimensional 共2D兲 Lamb–Oseen vortex with a heavy core predicts the existence of 2D temporally oscillating unstable modes.4 These unstable modes are of the Rayleigh–Taylor 共RT兲 type. As a matter of fact, heavy fluid inside light fluid in a vortex core is equivalent to light fluid below heavy fluid in the presence of gravity. The gravity force is replaced here by the centrifugal force. In both cases, the unstable situation corresponds to the force 共gravity or centrifugal兲 directed towards the light fluid. In this Letter, we study the linear and nonlinear development of these instabilities thanks to a 2D direct numerical simulation 共DNS兲. The main objective of this Letter is to characterize possible differences that may exist in the development of RT instabilities between the standard case with the gravity force and the present situation with the centrifugal force. Let Lref , uref , ␳ref and Tref be the reference length, velocity, density, and temperature scales of the problem. These reference scales will be precised below. The compressible Navier–Stokes equations read in nondimensional form

⳵t␳ + ⳵i共␳ui兲 = 0,

⳵ t共 ␳ u i兲 + ⳵ j 共 ␳ u iu j 兲 = − ⳵ i p +

共1兲

⳵ j␶ij , R

共2兲

Telephone: 33 共0兲1 46 23 51 64. Fax: 33 共0兲1 46 23 51 58. Electronic mail: [email protected]

a兲

1070-6631/2005/17共2兲/021703/4/$22.50

⳵tE + ⳵i关共E + p兲ui兴 =

⳵i共␶iju j兲 ⳵ i⳵ iT + , R 共␥ − 1兲PM2R

共3兲

where ui, ␳, T, p are the velocity, density, temperature, and pressure of the fluid, ␶ij = ⳵ jui + ⳵iu j − 2␦ij⳵kuk / 3 is the viscous stress tensor and E = p / 共␥ − 1兲 + ␳uiui / 2 is the total energy. The flow is supposed to behave like a perfect gas: ␥M2 p = ␳T. Several nondimensional parameters appear: the Reynolds number R = uref Lref ␳ref / ␮ where ␮ is the viscosity, the Mach number M = uref / 共␥RTref 兲1/2 where R is the constant of perfect gas, the specific heat ratio ␥, the Prandtl number P = ␮c p / k, where c p and k designate the specific heat at constant pressure and the thermal conductivity. The simulation is performed at a low Mach number so that the static temperature variations are small. This allows us to make the assumption of constant viscosity, constant thermal conductivity and constant specific heats. These equations are solved on a Cartesian grid mesh. Time integration of the equations is decoupled from spatial discretization and is perfomed by a third order Runge–Kutta scheme with the numerical coefficients of Lowery and Reynolds.5 The convective terms are discretized with a sixthorder compact finite differences scheme.6 To minimize the aliasing errors, the nonlinear terms are written in the skew symmetric form.7 Diffusive terms are evaluated with a sixthorder accurate scheme, except the diagonal terms that are discretized by using a scheme introducing enough numerical dissipation to avoid oscillations.8 Nonreflecting boundary conditions are prescribed at the lateral boundaries.9 The flow field is initialized by the superposition of a basic flow and a perturbation. The basic flow is an axisymmetric vortex with radial variations of the density profile. Hence, if ur and u␪ designate the radial and azimuthal velocity components in cylindrical 共r , ␪兲 coordinates, the basic flow reads: 关ur , u␪ , p , ␳兴 = 关0 , V共r兲 , P共r兲 , R共r兲兴. The velocity field V共r兲 is chosen to be the azimuthal velocity profile of the Lamb–Oseen vortex:

17, 021703-1

© 2005 American Institute of Physics

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Phys. Fluids 17, 021703 共2005兲

Coquart, Sipp, and Jacquin

FIG. 1. Modulus of the eigenfunctions ␳ˆ 共r兲, uˆr共r兲, and uˆ␪共r兲 associated to the eigenmode characterized by 共␴ = 0.20, ␻r = 2.82兲, m = 3, b = 0.3, s = 0.2, M = 0.1, ␥ = 1.4, R = 10 000, and P = 0.7. FIG. 2. Computational domain and grid mesh.

V=

1 − exp共− r2兲 . r

共4兲

The reference length and velocity scales Lref and uref therefore designate the characteristic radius and velocity of the Lamb–Oseen vortex. In addition, for the definition of the Mach number M, we choose the static temperature of the fluid at infinity as the reference temperature scale Tref . The density distribution R共r兲 is prescribed as R = 1 + s exp共− r2/b2兲,

共5兲

where s and b designate the amplitude and width of the heavy internal core. The reference density scale ␳ref is therefore equal to the density at infinity. The pressure P共r兲 is obtained by integrating the radial momentum equation: P=

1 − ␥M2

冕

⬁

r

RV2 dr. r

共6兲

This flow field is a steady solution of the Euler equations while it slightly diffuses in time in the presence of viscosity. The perturbation is constituted of a small amplitude eigenmode of the linearized Navier–Stokes equations around the basic flow 共4兲–共6兲. This eigenmode is calculated thanks to a matrix eigenvalue–eigenvector method based on a spectral Chebyshev collocation. If Re designates the real part of a complex, the unstable eigenmode may be written as

关ur⬘,u␪⬘,p⬘, ␳⬘,T⬘兴 = Re共A关uˆr共r兲,uˆ␪共r兲,pˆ共r兲, ␳ˆ 共r兲,Tˆ共r兲兴ei共m␪−␻t兲兲,

共7兲

where m is the azimuthal wavenumber, ␻ is the complex frequency ␻ = ␻r + i␴, and A is the amplitude of the eigenmode. A Lamb–Oseen vortex with a heavy internal core 共s ⬎ 0兲 is generically unstable to RT instabilities. The involved physical mechanims are incompressible and inviscid. In the following, we therefore choose a low value for the Mach number and a high value for the Reynolds number. More precisely, the case investigated in this Letter corresponds to b = 0.3, s = 0.2, M = 0.1, ␥ = 1.4, R = 10 000, and P = 0.7. It appears that the most unstable eigenmode in this situation corresponds to an m = 3 eigenmode with 共␴ = 0.20, ␻r = 2.82兲. The associated eigenfunctions ␳ˆ 共r兲, uˆr共r兲 and uˆ␪共r兲 are represented in Fig. 1. For the initialization of the simulation, the amplitude A of the eigenmode in Eq. 共7兲 is adjusted so as to have max兵␳⬘其 = 0.02max兵R其. The computational domain is represented in Fig. 2. Its dimensions are: −30艋 x, y 艋 30. The grid is uniform in the domain −3 艋 x, y 艋 3 and is stretched using a monotonic tanh law up to the boundaries. The mesh has Nx = 681 and Ny = 681 points. The density peak is described with 30 points and the core of the vortex with 100 points.10 The time step is ⌬t = 5 · 10−4, which corresponds to CFL= 0.55. At each time step, we evaluate the mean density and azimuthal velocity profiles by averaging over ␪. For instance,

FIG. 3. Evolution of 共a兲 ¯␳共r , t兲 and 共b兲 ¯u␪共r , t兲 as a function of the radius r for t = 0.157; t = 10.969; t = 14.103; t = 20.37; t = 34.47.

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Mixing induced by Rayleigh–Taylor instability

FIG. 4. Amplitude of the perturbation ␳⬘2共t兲 as a function of time t.

¯␳共r , t兲 = 共1 / 2␲兲兰20␲␳共r , ␪ , t兲d␪. Figures 3共a兲 and 3共b兲 represent these two quantities, i.e., ¯␳共r , t兲 and ¯u␪共r , t兲 as functions of the radius r at various times t. We can see that the Gaussian density peak collapses as time evolves while the azimuthal velocity field of the vortex remains nearly unchanged. Hence, as the instability develops, the nonlinear interactions destroy the density peak of the mean flow, which renders the flow stable to RT instabilities. The density fluctuations are then evaluated following ␳⬘共r , ␪ , t兲 = ␳共r , ␪ , t兲 − ¯␳共r , t兲. The overall amplitude of the perturbation may be obtained from an integral over the calculation box of the square of the density fluctuations: ␳⬘2共t兲 = 兰⬁0 兰20␲关␳⬘共r , ␪ , t兲兴2rdrd␪. Figure 4 shows this quantity as a function of time t. Its growth rate may be compared to

Phys. Fluids 17, 021703 共2005兲

FIG. 5. Density fluctuations ␳⬘共r , ␪ , t兲 at 共x = 0.289, y = 0兲 as a function of time t.

the amplification rate obtained from the linear stability analysis. Agreement is very good in the linear phase, i.e., up to t ⯝ 5. Then, the mean amplitude of the perturbation saturates at t ⯝ 10 before decreasing slowly with time. For t ⬎ 20, as shown before, the density peak of the mean flow collapses as time evolves. Hence, the driving force of the instability vanishes which induces the loss of the perturbation source. The density fluctuations therefore die out and the vortex flow becomes laminar. Figure 5 shows the density fluctuations ␳⬘共r , ␪ , t兲 obtained at 共x = 0.289, y = 0兲 as a function of time t. In the linear regime, i.e., for t 艋 5, the signal exhibits a time period equal to ⌬t = 2.2. This value is compatible with the linear stability

FIG. 6. 共Color兲. Density ␳共r , ␪ , t兲 at different times: 共a兲 t = 0.157; 共b兲 t = 10.969; 共c兲 t = 20.37; 共d兲 t = 34.47.

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Phys. Fluids 17, 021703 共2005兲

Coquart, Sipp, and Jacquin

analysis which predicts a time period equal to ⌬t = 2␲ / ␻r = 2.2. In the nonlinear regime, the signal becomes more abrupt and decreases in amplitude. Figure 6 presents density isovalues at four distinct times. In Fig. 6共a兲, the RT instability starts growing linearly. In Fig. 6共b兲, we enter the nonlinear regime: the flow exhibits the mushroom-like patterns characteristic of the nonlinear development of the RT instability.11 In Fig. 6共c兲, which corresponds to the late stage development of the instability, we observe nonstandard features. Usually, i.e., in the case of standard nonhomogeneous flows with gravity, the RT mushroom-like patterns just continue to spread while new secondary instabilities complexify the overall flow structure. In the case of a vortex with a heavy internal core, the mushroom-like patterns are strongly stretched in the azimuthal direction and subject to filamentation.These differences are due to the azimuthal velocity of the vortex which is not present in the standard case. Comparing Fig. 6共b兲 to Fig. 6共c兲, it can also be observed that the intensity of the perturbations decreases with time. As shown in Fig. 6共d兲, a complex lamellar structure is finally obtained in the vortex core. To conclude, we have performed a 2D DNS of the RT instability which develops in a vortex with a heavy internal core. The results show that after a phase of linear growth, the amplitude of the perturbation saturates and finally decreases. The nonlinear interactions destroy the density peak of the mean flow, which is precisely the driving force of the instability. Hence, the mean flow becomes stable to RT instabilities, which leads to the loss of the perturbation source. Also, in the late stage development of the nonlinear regime, we have shown that the azimuthal velocity of the vortex violently stretches the perturbations in the azimuthal direction

leading to a strong radial filamentation of the flow. Hence, the nonlinear evolution of the RT instabilities obtained here is radically different from the standard case of RT instabilities developing in a planar nonhomogeneous flow in the presence of gravity. This shows that differential rotation may deeply affect the development of RT instabilities. Considering practical applications, the potential of density effects on the momentum diffusion of a vortex is poor. But, the mechanisms described above also show that one obtains very efficient mixing which remains localized within the vortex core. These properties may have interesting applications in the field of mixing and combustion. 1

O. Zeman, “The persistence of trailing vortices: A modeling study,” Phys. Fluids 7, 135 共1995兲. 2 L. Jacquin and C. Pantano, “On the persistence of trailing vortices,” J. Fluid Mech. 471, 159 共2002兲. 3 P. R. Spalart, “Airplane trailing vortices,” Annu. Rev. Fluid Mech. 30, 107 共1998兲. 4 D. Sipp, D. Fabre, S. Michelin, and L. Jacquin, “Stability of a vortex with a heavy core,” J. Fluid Mech. 共to be published兲. 5 P. S. Lowery and W. C. Reynolds, “Numerical simulation of spatially developing forced plane mixing layer,” Stanford University, Report No. TF-26, 1986. 6 S. Lele, “Compact finite differences schemes with spectral resolution,” J. Comput. Phys. 103, 16 共1992兲. 7 G. A. Blaisdell, E. T. Spyropoulos, and J. H. Qin, “The effect of the formulation of nonlinear terms on aliasing errors in spectral methods,” Appl. Numer. Math. 21, 207 共1996兲. 8 C. F. Gago, S. Brunet, and F. Garnier, “Numerical investigation of turbulent mixing in a jet/wake vortex interaction,” AIAA J. 40, 276 共2002兲. 9 K. W. Thompson, “Time dependent boundary conditions for hyperbolic systems,” J. Comput. Phys. 68, 1 共1987兲. 10 M. Sreedhar and S. Ragab, “Large eddy simulation of longitudinal stationary vortices,” Phys. Fluids 6, 2501 共1994兲. 11 T. T. Clark, “A numerical study of the statistics of a two-dimensional Rayleigh–Taylor mixing layer,” Phys. Fluids 15, 2413 共2003兲.

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